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In the case of Cos[x] – Cos[α x] each step in the generalized substitution system has a rule determined as shown on the left from a term in the continued fraction representation of (α–1)/(α+1) .

[Mollusc] shell model
The center of the opening of a shell is taken to trace out a helix whose {x, y, z} coordinates are given as a function of the total angle of revolution t by a t {Cos[t], Sin[t], b} . … The complete surface of the shell is obtained by varying both t and θ in
a t {Cos[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), Sin[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), b + c (Cos[ θ ] Sin[e] - d Cos[e] Sin[ θ ])}
where c varies from 0.4 to 1.6 on row (c), d from 1 to 4 on row (d) and e from 0 to 1.2 on row (e).

In the specific case a = 2 the iterates of If[x < 1/2, a x, a (1 - x)] have the form ArcCos[Cos[2 n π x]]/ π .

Intrinsically defined curves
With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 )
NDSolve[{x'[s] Cos[ θ [s]], y'[s] Sin[ θ [s]], θ '[s] f[s], x[0] y[0] θ [0] 0}, {x, y, θ }, {s, 0, s max }]
For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve :
f[s] = 1: {Sin[ θ ], Cos[ θ ]}
f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]}
f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]}
f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]}
f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]}
f[s] = s n : result involves Gamma[1/n, ± θ n/n ]
f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2,
Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3])
Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2
Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}]
(As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .) And in this way the discrete system x If[EvenQ[x], 3x/2, 3(x + 1)/2] from page 122 can be emulated by the continuous iterated map x (3 + 6 x - 3 Cos[ π x])/4 .

Substitution systems [and sine sums]
Cos[a x] - Cos[b x] has two families of zeros: 2 π n/(a + b) and 2 π n/(b - a) .

In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms
Sin[2 t ArcSin[ √ x ]] 2
(1 - Cos[2 t ArcCos[1 - 2 x]])/2
(1 - ChebyshevT[2 t , 1 - 2x])/2
where these follow from functional relations such as
Sin[2x] 2 4 Sin[x] 2 (1 - Sin[x] 2 )
ChebyshevT[m n, x] ChebyshevT[m, ChebyshevT[n, x]]
For a = 2 it also turns out that there is a complete formula:
(1 - (1 - 2 x) 2 t )/2
And the same is true for a = -2 :
1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))]
In all these examples t enters essentially only in a t . … When a = 2 it is Exp[x] and when a = -2 it is 2 Cos[(1/3) ( π - √ 3 x)] .

Two sine functions
Sin[a x] + Sin[b x] can be rewritten as 2 Sin[1/2(a + b) x] Cos[1/2(a - b) x] (using TrigFactor ), implying that the function has two families of equally spaced zeros: 2 π n/(a + b) and 2 π (n + 1/2)/(b - a) .

[Spectra of] random block sequences
Analytical forms for all but the last spectrum are: 1 , u 2 /(1 + 8u 2 ) , 1/(1 + 8 u 2 ) , u 2 , (1 - 4u 2 ) 2 /(1 - 5u 2 + 8u 4 ) , u 2 /(1 - 5u 2 + 8u 4 ) , u 2 + 1/36 DiracDelta[ ω - 1/3] , where u = Cos[ π ω ] , and ω runs from 0 to 1/2 in each plot. … MatrixPower[ m[Map[Length, list]], r] . w/Length[w]]
then forming Sum[ ξ [Abs[r]] Cos[2 π r ω ], {r, -n/2, n/2}] and taking the limit n ∞ . If ξ [r] = λ r then the spectrum is (1 - λ 2 )/( λ 2 - 2 λ Cos[2 π ω ] + 1) - 1 .

In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.